Methods for multi-objective optimization using evolutionary algorithms

Abstract

In the field of multi-objective optimization using evolutionary algorithms conventionally different objectives are aggregated and combined into one objective function using a fixed weight when more than one objective needs to be optimized. With such a weighted aggregation, only one solution can be obtained in one run. Therefore, according to the present invention two methods to change the weights systematically and dynamically during the evolutionary optimization are proposed. One method is to assign uniformly distributed weight to each individual in the population of the evolutionary algorithm. The other method is to change the weight periodically when the evolution proceeds. In this way a full set of Pareto solutions can be obtained in one single run.

Description

[0001]This application is related to U.S. application Ser. No. 10 / 007,734, filed on Nov. 9, 2001 by Y. Jin and Dr. B. Sendhoff, titled “Approximate Fitness Function” and is related to U.S. application Ser. No. 10 / 501,378, is related to U.S. application Ser. No. 11 / 501,573 filed on Aug. 8, 2006, and is related to U.S. Pat. No. 7,243,056, and is related to U.S. application Ser. No. 11 / 042,991, and is related to U.S. application Ser. No. 11 / 033,767, and is related to U.S. application Ser. No. 11 / 517,135.[0002]The present invention relates to a method for the optimization of multi-objective problems using evolutionary algorithms as well as to a computer software program product for implementing such a method.[0003]The background of the present invention thereby is the field of evolution strategy. Therefore with reference to FIG. 1 at first the known cycle of an evolutionary algorithm will be explained.[0004]In a step 1 the object parameters to be optimized are encoded in a string called...

AI Technical Summary

Benefits of technology

[0025]Despite their weaknesses, the aggregation approaches are very easy to implement and computationally efficient. Usually, aggregation approaches can provide only one Pareto-solution if the weights are fixed using problem-specific prior knowledge. However, it is also possible to find more than one Pareto solutions using this method by changing the weights during optimization. The weights of the different objectives are encoded in the chromosome to obtain more than one Pareto solutions. Phenotypic fitness sharing is used to keep the diversity of the weight combinations and mating restrictions are required so that the algorithm can work properly.

[0027]Therefore, it is the target of the present invention to develop a technique for a multi-objective optimization using evolutionary algorithms with a reduced computational effort.

[0029]To approximate the Pareto front instead of a single Pareto solution, the weight for each objective is changed systematically. One method is to distribute the weights uniformly among the individuals in the population. A further method is to periodically change the weights with the process of the evolution. Although these methods seem to be very simple, they have proved to be working effectively for two-objective optimization problems. Both methods work well on low-dimensional problems. However, for high-dimensional problems, the second method outperforms the first one. On the other hand, it also depends on the performance of the evolution strategy. The evolution strategy with Rotation Matrix Adaptation can give better performance than the standard evolution strategy. At the same time, the evolution strategy with Covariance Matrix Adaptation provides very good results on smooth, high-dimensional problems, but its performance degrades seriously on problems with discontinuous and non-convex Pareto-optimal front.

Problems solved by technology

On the contrary, in a multi-criterion optimization with conflicting objectives, there is no single optimal solution.

In dealing with multi-criterion optimization problems, classical search and optimization methods are not efficient, simply because (i) most of them cannot find multiple solutions in a single run, thereby requiring them to be applied as many times as the number of desired Pareto-optimal solutions, (ii) multiple application of these methods do not guarantee finding widely different Pareto-optimal solutions, and (iii) most of them cannot efficiently handle problems with discrete variables and problems having multiple optimal solutions.

Unfortunately, the Pareto-based approaches are often very time-consuming.

However, for high-dimensional problems, the second method outperforms the first one.

At the same time, the evolution strategy with Covariance Matrix Adaptation provides very good results on smooth, high-dimensional problems, but its performance degrades seriously on problems with discontinuous and non-convex Pareto-optimal front.

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